In Chapter IX, Sec 6, Ex 3 of CWM, MacLane defines the twisted arrow category of C such that objects are arrows f : a -> b of C and arrows are pairs of morphisms of C, (l,m) : f -> g, such that g = mfl. The construction is part of a proof of the reduction of ends to limits.
I would be grateful for pointers to other occurrences of this construction in the literature.
The twisted arrow category of A is the category of elements of the hom functor A(-,-) : A^op x A --> Set. In internal and variable category theory there is a sense in which the twisted arrow fibration comes first and then can be used to define hom functors and local smallness. For example, see page 292 of my paper Cosmoi of internal categories, Transactions American Math. Soc. 258 (1980) 271-318; MR82a:18007 Regards, Ross www.mpce.mq.edu.au/~street/